integration - Finding the integral $int_0^pifrac{dtheta}{(2+costheta)^2}$ by complex analysis

Trying to find the integral $$\int_0^\pi\frac{d\theta}{(2+\cos\theta)^2}$$ by complex analysis.

I let $z = \exp(i\theta)$, $dz = i \exp(i\theta)d\theta$, so $ d\theta=\dfrac{dz}{iz}$. I am trying therefore to find the integral $$\frac{1}{2iz} \oint_C \frac{dz}{\left(2 + \frac{z}{2} + \frac{1}{2z}\right)^2}$$ I am unsure of which contour I should use, and how to proceed besides that. Could anyone help?